Intermediate value theorem

Intuition

A continuous function cannot move from one value to another without taking every intermediate value along the way.
For example, if you are hiking, and your altitude (a continuous function of time) starts at $1000$ and ends at $2000$, then you can be sure that at some point during the hike, the altitude will be exactly $1257$.

Statement

Let $f:[a,b]\rightarrow \R$ be a continuous function on the closed interval $[a,b]$. If $L$ is any real number between $f(a)$ and $f(b)$, then there exists at least one $c$ in $[a,b]$ such that
$$f(c)=L.$$.